4. Potential Flow Theory

(From Aerodynamics for Students Website)

We consider only inviscid flows in this chapter. This leads to a simple analysis. In fact
we will be solving only the continuity equation for mass to calculate
velocity components. Pressure is obtained from Bernoulli equation. Of
course, various assumptions will have to made to make the analysis easier.
We discuss these in course of the text.

We first derive the continuity equation which is a statement of the
fact that mass is conserved. Then we introduce the stream function which
is a powerful concept in fluid dynamics. By analyzing the kinematics of
fluid motion we proceed to introduce concepts of Circulation and
Irrotationality. Definition of Velocity Potential
follows. We then write down the stream functions and velocity potentials
for some of the simple flows like a uniform flow, source and sink flow and
vortex flow. These flows are then superposed to arrive at solutions for
complicated flows. Flow about a circular cylinder is then analyzed in some
detail.

We derive the equation for mass conservation by considering a
differential control volume at P(x,y,z) as shown
in Fig.4.1. Let
the dimensions of the volume be dx, dy and dz and velocity
components at P be u,
v and w. Assuming that the mass flow rate
is continuous across the volume we can calculate the mass flow rates at
the various faces of the cell by a Taylor Series expansion as we had done
previously (Eqn. 2.5).
Accordingly we have,

(4.1)

The net mass flow rate into the control volume as a consequence is
given by,

Further in Eqn.4.3 noting
that the control volume is tiny, the integral can be approximated as

(4.5)

The Reynolds Transport Theorem thus gives,

(4.6)

Cancelling out dx dy dz, we have,

(4.7)

Eqn. 4.7 is
known as the Continuity Equation. Note that it
is a very general equation with hardly any assumption except that density
and velocities vary continually across the element we have considered.

If we now bring in the gradient operator, namely,

(4.8)

and represent velocity as a vector,

(4.9)

Then the Continuity Equation can be written in a compact manner as

(4.10)

Written in this form it enables one to consider any other system of
coordinates with ease.

We have derived the Continuity Equation, 4.10
using Cartesian Coordinates. It is possible to use the same system for all
flows. But sometimes the equations may become cumbersome. So depending
upon the flow geometry it is better to choose an appropriate system. Many
flows which involve rotation or radial motion are best described in
Cylindrical Polar Coordinates. Let us now write equations for such a
system. In this system coordinates for a point P are and , which
are indicated in Fig.4.2. The
velocity components in these directions respectively are and .
Transformation between the Cartesian and the polar systems is provided by
the relations,

Stream function is a very useful device in the study of fluid dynamics
and was arrived at by the French mathematician Joseph
Louis Lagrange in 1781. Of course, it is related to the streamlines
of flow, a relationship which we will bring out later. We can define
stream functions for both two and three dimensional flows. The latter one
is quite complicated and not necessary for our purposes. We restrict
ourselves to two-dimensional flows.

Consider a two-dimensional incompressible flow for which the continuity
equation is given by,

Thus the continuity equation is automatically satisfied. Thus if we can
find a stream function that meets with the
eqn.4.19 the
continuity equation need not be solved. For the rest of the chapter we
will be invariably describing flows with a stream function.

Consider a two-dimensional fluid element, a square ABCD for simplicity. when the fluid flows this element
is subject to various forces and as a result undergoes a complex motion
and a possible deformation as indicated in Fig.4.5 and
assumes a shape like A`B`C`D`. It appears that
the complex deformation of the element can be split into four basic
constituents -

Translation

Linear Deformation

Rotation

Angular Deformation

Figure 4.5 :Deformation
of A fluid Element

Figure 4.6 : Basic Deformation of A
fluid Element

These elemental deformations have been sketched in Fig.4.6 .
Let us now consider each of these separately.

Translation is the type of motion where the element retains its shape.
Its side do not undergo any change in length and the four angles do remain
square. The square element ABCD, we
have considered is bodily shifted from its position to a new one A`B`C`D`. If the motion is without
acceleration in a steady, uniform flow it is easy to calculate the
position of any particle in the fluid at different instants of time. But
it should be noted that the fluid particles may undergo acceleration.

Figure 4.7: Translation
of A fluid Element

Considering now a particle in the fluid element we can write
down an expression for its acceleration. At time t, the particle is at (x,y,z) and its velocity is given by,

(4.25)

where is the fluid velocity. At time (t+dt) the particle will find itself at a different
location and its velocity now is given by,

(4.26)

Consequently the change in velocity is given by,

(4.27)

The acceleration of the particle is obtained by dividing throughout by
dt. Accordingly,

(4.28)

Now denoting the speed in x-direction, dx/dt by u, speed in y-direction dy/dt by v and speed in z-direction
dz/dt by w, we have,

(4.29)

where ap is the particle
acceleration. The derivative dVp/dt
is usually denoted by DV/Dt and is called Material Derivative or the Particle
Derivative or the Total Derivative.

Consider the same element ABCD again. If the
element has to undergo a linear deformation it is necessary that u velocity change in the x-direction and v velocity
in the y-direction. Let the velocities at A be u and v. Then at B the u-velocity will be
and at D it will be . As a result the element stretches both in x and y
directions and assumes a shape A`B`C`D` shown in
Fig. 4.8.

Figure 4.8: Linear
Deformation of a Fluid Element

Now, the stretch in x-direction is given by distance BB' which is equal to

(4.30)

Corresponding change in the volume of the element (for a unit depth
normal to the paper) is given by

(4.31)

Similarly the change in the volume of the element in the
y-direction is given by,

(4.32)

Neglecting the change in volume due to CC```C`C``, The total change in volume is

(4.33)

Thus the rate of change of volume expressed as a fraction of the
initial volume is given by,

(4.34)

The left hand side is called the Volume Dilitation
rate of the element. We have seen that the Right hand side of the
equation is zero for all incompressible flows.

Considering the same element ABCD again, we
notice that any rotation of AB or AD is brought about by a change in u velocity along y-direction
and that of v velocity along x-direction. Let and
be the angles through which sides AB and AD rotate.

Figure 4.9 : Rotation of
a fluid Element

Now it is required that

(4.35)

Since angles
and
are small we have,

(4.36)

Rotation or angular
velocity of the element (about the z-axis) is defined as

It should be noted that the concept of Irrotationality applies to a
fluid element in a given flow than to the flow itself. The main flow may
be a vortex where the streamlines are circles. But the individual elements
of fluid may not rotate or distort making the flow irrotational. This is
shown in Fig. 4.10
where an irrotational flow about an aerofoil is sketched. Note that even
though the main follows a path which seems to indicate distortion, the
fluid elements are simply translated.

We now discuss one other property of flows, that of Circulation. Consider any closed curve C in a flow as shown. Circulation is defined as the
line integral around the curve of the arc length
ds times the tangential component of velocity. Shear stress for the
element is thus given by

Figure 4.11 :
Definition of
Circulation

(4.45)

An expression for circulation can be derived by considering a small
differential area in the curve, which is shown enlarged in Fig.4.12.

Figure 4.12:
Definition of Circulation, continued

If the velocity components at A are (u,v) then we have

Velocity along AB:

Velocity along BC:

Velocity along DC:

Velocity along AD:

(4.46)

Circulation along the boundary of the differential element is given by

We have seen that for an irrotational flow . It follows from vector algebra that there should be a
potential such that

(4.53)

is called the Velocity Potential.
The velocity components are related to through the
following relations.

(4.54)

Velocity potential is a powerful tool in analysing irrotational flows.
First of all it meets with the irrotationality condition readily. In fact,
it follows from that condition. As a check we substitute the velocity
potential in the irrotationality condition, thus,

(4.55)

The next question we ask is does the velocity potential satisfy the
continuity equation? To find out we consider the continuity equation for
incompressible flows and substitute the expressions for velocity
coordinates in them. Accordingly,

(4.56)

It is clear that to meet with the continuity requirements the velocity
potential has to satisfy the equation,

(4.57)

In vector notation it is

(4.58)

As with stream functions we can have lines along which potential is constant. These are called Equipotential Lines of the flow. Thus along a
potential line .

The equation 4.58 is
called the Laplace Equation and is encountered
in many branches of physics and engineering. A flow governed by this
equation is called a Potential Flow. Further the
Laplace equation is linear and is easily solved by many available standard
techniques, of course, subject to boundary conditions at the boundaries.

Note that in terms of velocity potential expression for
circulation(Eqn. 4.45, see
Circulation) assumes a simple form.

We notice that velocity potential and stream function
are connected with velocity components. It is necessary
to bring out the similarities and differences between them.

Stream function is defined in order that it satisfies the continuity
equation readily (eqn. 4.20 see
Stream function). We do not know yet
if it satisfies the irrotationality condition. So we test out below.
Recall that the velocity components are given by

Substituting these in the irrotationality condition, we have

(4.61)

Which leads to the condition that for irrotationality.

Thus we see that the velocity potential
automatically complies with the irrtotatioanlity condition, but to satisfy
the continuity equation it has to obey that . On the other hand the stream
function readily satisfies the continuity condition, but to meet with the
irrotationality condition it has to obey .

Thus we see that the streamlines too follow the Laplace Equation. So it
is possible to solve for a potential flow in terms of stream function.

Property

Continuity Equation

Automatically Satisfied

satisfied if =0

Irrotationality Condition

satisfied if =0

Automatically Satisfied

Table 4.1 : Properties
of stream function and velocity potential

Streamlines and equipotential lines are orthogonal to each other. We
have seen that the velocity components of the flow are given in terms of
velocity potential and stream function by the equations,

(4.62)

Those familiar with Complex Variables theory will recognise that these
are the Cauchy-Riemann equations and that and are orthogonal and that both and obey Laplace
Equation. However, we will prove the orthogonality condition by other
means.

Figure 4.13 : Orthogonality of Stream lines and Equi-potential lines

Since , it follows that

¡@

¡@

(4.63)

The gradient of the equipotential line is hence given by

(4.64)

On the other hand the gradient of a stream line is given by

(4.65)

Thus we find that

(4.66)

showing that equipotential lines and streamlines are orthogonal to
each other. This enables one to calculate the stream function when the
velocity potential is given and vice versa.

Fig. 4.14
shows the flow through a bend where the streamlines and the equipotential
lines have been plotted. The two form an orthogonal network.

Figure 4.14 : Stream lines and
Equi-potential lines for flow through a bend

A question that naturally arises is "Where do we find irrotational
flows?". A uniform flow is definitely irrotational. But one hardly finds a
uniform flow in nature. Further, there is hardly anything to calculate for
a uniform flow.

The other region where we can expect an irrotational flow is away from
any solid body. Recall the "Thought Experiment" with two parallel plates
(What is a
Fluid?) when the space in between is filled with a fluid. Here once
the top plate starts moving we have seen that a velocity gradient is set
up in the flow normal direction. This gives rise to which contributes directly to
vorticity or rotation. As such this flow is NOT
irrotational. A similar velocity gradient is set up when a fluid flows
past a solid body as shown in Fig.4.15.
The velocity right on the body surface is zero and it build up gradually
we move in a normal direction away from the body. This region is highly
rotational and is called the Boundary Layer. But at some distance form the
body this velocity gradient flattens out and the velocity becomes constant
in the flow normal direction. This is one of the irrotational regions of
flow. As indicated in the figure the flow in the wake of the body is also
NOT irrotational.

Figure 4.15:
Occurrence of irrotational and rotational regions for
flow past a body

Figure 4.16:
Occurrence of
irrotational and rotational regions for flow through a pipe.

At the entrance to a pipe as shown in Fig.4.16
one has a uniform flow. As the flow enters the pipe, velocity components
are forced to be zero on the surface of the pipe. A boundary layer
develops and starts to grow. At the beginning one sees a inviscid core
encircled by a boundary layer. The flow in the inviscid core is
irrotational. However, as we move downstream the boundary layers grow and
merge to give a fully developed flow when the entire flow is NOT irrotational.

It is also worth noting that the flow is irrotational wherever
Bernoulli equation is valid.

We could foresee from this that an inviscid flow is likely to be
irrotational. In fact it is broadly true except in case of High Speed
flows where shocks could occur. As indicated in Fig. the region behind a
shock in a high speed flow has severe gradients of velocity making
not negligible.

In this section we consider some of the simple potential flows. The
examples considered are such that there is an analytical expression for for each of them. While calculating such
flow a good coordinate system is important. Of course, it is true that any
example could be handled with the Cartesian coordinates. But depending on
the problem, the expressions may become too complicated. Further, the
geometry of the flow itself indicates the coordinates to be chosen.
Accordingly it is necessary to write down the important formulas involving
stream function and velocity potential in the Cartesian and Polar
coordinate systems.

Consider a radial flow going away from the origin at a velocity as shown in Fig.4.19.
This constitutes a Source Flow. This is a purely radial flow with no
component of velocity in the tangential direction, i.e., . If m is
the volumetric flow rate we have

Figure 4.19 : Source Flow and Sink
Flow.

i.e.,

(4.74)

We can now write down velocity potential and stream function for this
flow :

(4.75)

It is easily verified that for this flow. Further, the equation we started out with
, namely, Eqn.4.74 is
the continuity equation for the source flow. It states that the Volumetric
flow rate (mass flow rate when multiplied by density) is constant in a
radial direction and is equal to m, which is
called the Strength of the source.

Another point to make is that the radial velocity becomes infinite at r = 0. So the origin is a singularity of the
flow.

If m is negative we have a flow which flows
inwards and is called a Sink flow, which again
has a singularity at the origin.

We now consider flows which go in a circumferential direction with no
radial flow. These are Vortex flows as shown in Fig. 4.20.

Figure 4.20: A Vortex Flow

The velocity potential and stream function are given by,

(4.76)

The velocity components are given by

(4.77)

It is seen that is infinite at the origin and decreases as r
increases and becomes zero as r approaches
infinity.

A question arises now as to whether we are contradicting ourselves? How
is it that a vortex flow is irrotational? We should note that the term
"Irrotational" refers to the behaviour of a fluid element and not to the
path taken by it. At an elemental level the flow is still irrotational.
Such a vortex is called a Free Vortex. A good
and familiar example is that of a bath tub vortex. Contrary to this we
have a Forced Vortex which behaves like a solid
body. These have their velocity given by
, with a zero velocity at the origin. The velocity
increases as one moves away from the origin. A water filled tank is a good
example.

Let us now calculate the circulation (see Circulation)
around a free vortex. We have

(4.78)

which is non-zero. Where is the flaw in our integration then? It is a
simple matter to find out. We have a singularity in our region, namely,
r = 0! If we exclude the singularity by making a
small cut around the origin, we will in fact get the result that
circulation around the vortex is zero.

It is usual to write the equation for velocity potential and stream
function in terms of circulation , thus

A Doublet is
formed when the source and sink approach each other,i.e., and at the same time such that is constant, we see that

(4.86)

As a consequence the stream function becomes

(4.87)

The velocity potential is

(4.88)

Figure 4.23: Doublet Flow

The streamlines and the equipotential lines for a doublet are sketched
in Fig.4.23. It
is seen that the streamlines are circles which are tangential to the x-axis while the equipotential lines are also circles
but tangential to y-axis.

One question that naturally arises is "Why are we discussing these
flows such as uniform flow, source flow vortex and doublet flows when they
do not actually exist. The answer is "Yes, they do not exist. But
conceptually they are useful.". In fact, they serve as alphabets of
potential flow. By combining these flows we can build up more complicated
flows, which are meaningful.

We note that the flows we have discussed are linear flows. By linearity
it is meant that if A and B are two solutions, even (where m and n are numbers) is also a solution. Therefore we are
allowed to superpose the elementary flows one above the other to obtain a
different flow. If the constituent flows are irrotational, the combined
flow too is irrotational.

The student is advised to run the following programs which plot
streamlines and velocity potential lines for various cases of
superposition.

Let us now place a source in the path of a uniform flow. The stream
function and the velocity potential for the resulting flow are given by
adding the two stream functions and velocity potentials as follows,

¡@

(4.89)

¡@

(4.90)

One of the interesting features to determine for the resulting force is
the stagnation point of the flow, i.e., where the velocity goes to zero.

One could calculate this from the equations. It is clear that for this
flow the stagnation point will occur on the x-axis. The location can be arrived at purely
intuitionally. The source produces a radial flow of magnitude

while the uniform flow produces a velocity of U in the positive
x-direction. When these two cancel out at a point we have the
stagnation point. A negative radial flow that can cancel the uniform flow
is possible only to the left of the x-axis, say
at x = -b. Hence,

leading to

(4.91)

At x = -b, we have and r = b.
Substituting these values in the expression for , i.e.,
Eqn. 4.89 we
get the value of at the stagnation
point to be

(4.92)

An equation to the streamline passing through the stagnation point,
i.e., stagnation streamline is obtained as follows,

But hence

leading to

(4.93)

Figure 4.24 : Flow about Rankine Half
Body

The streamlines for this flow are sketched in Fig.4.24. It
is clear that we can make the stagnation streamline the solid body. In
fact any streamline of a flow can be treated as a solid body since there
is no flow across it. In the present example if we ignore the streamlines
inside the "body" we have described the flow about a solid body given by
Eqn. 4.25.
This body is referred to as a Rankine Half Body
as it is "open" at the right hand end.

Limits of for this body are 0 and . At these values we have y
approaching , which is called the Half Width of the body.

which enables us to calculate the pressure. Usually in aerodynamic
applications involving significant velocities and pressures any
contribution due to elevation changes is negligible. The equation for
pressure assumes a simple form,

We saw that the previous example defined a half body open at one end.
Can we come up with a closed body by a suitable combination? An inspection
of the streamlines suggests that by placing a sink in addition to the
source one should be able to define a closed body. In other words a
uniform flow past a source-sink combination is what we are after. The
stream function for this is given by

When the streamlines for this flow are plotted (Fig.4.26) one
discovers that the one given by (shown in red) forms a closed curve. This
obviously forms the "body", i.e., the stream function we have written
describes the flow about this body. Shapes such as this are called Rankine Ovals. The distance to the stagnation points
from the origin or the Half Body Length is given by

(4.101)

Figure 4.27 : Rankine Oval

The other feature of interest, Half Width is found by determining the
point of intersection of y-axis with the body, i.e., line. An expression for h is,

(4.102)

the solution for which is to be obtained by iteration. Rankine ovals
include a wide range of bodies which can be obtained by varying the value
of the parameter . These could be bodies stretched in
any of the two directions. When stretched in
x-direction one obtains elliptic bodies with a small half width
compared to the span. The solution obtained could be a good approximation
to the flow especially if viscous effects are small. On the other hand a
considerable half width would indicate a bluff body prone to effects like
separation. The solution obtained can hardly be accepted in this case.

Flow around a circular cylinder can be approached from the previous
example by bringing the source and the sink closer. Then we are
considering a uniform flow in combination with a doublet. The stream
function and the velocity potential for this flow are given by,

has a zero at 0
and 1800 and a maximum of 1 at
= 900 and 2700. The former set denotes the stagnation
points of the flow and the later one denotes the points of maximum surface
velocity (of magnitude ). Thus the velocity decreases from a value of at equals 900 to as one moves away in a normal direction s shown in Fig 4.30.

The surface pressure distribution is calculated from Bernoulli
equation. If we denote the free stream speed and pressure as and we have

Fig. 4.31
shows Cp plotted as a function of
. A symmetry about y -axis is apparent. When compared to
the experimentally observed Cp
distribution we see that there is some agreement in the region
between= 00 and = 900 . But any agreement is lost in the other
regions. The reasons for this are obvious. Viscous forces dominate the
flow in the region to the right of the centreline giving rise to
separation. The pressure tends to plateau out in a separated region, the
level depending on whether it is a laminar separation or a turbulent
one.

Symmetry in the theoretical Cp
distribution about both y-axis and x-axis shows that drag and lift forces about the
cylinder are each zero. This may also be proved by integrating pressure
around the cylinder, thus,

What we have just calculated is in contrast to the experimental results
which do predict a significant drag for the flow about a circular
cylinder. This seems to have caused in what is called D'Alembert's Paradox in honour of Jean le Rond D'Alembert (1717-1783). Now it is no more
a paradox. As we discussed above we calculate a zero drag because we have
not taken viscosity into account.

The stagnation points we saw in Fig. 4.33 are
for the case when the circulation imposed on the cylinder was such that . But from Eqn. 4.131 it
is evident that angle , hence the position of the stagnation points is a strong
function of circulation, . This is illustrated in Fig.4.34. With
zero circulation the stagnation points lie at = 0, . As circulation increases the stagnation points move (upwards or
downwards depending upon the direction of rotation). When they coincide at = or = - . If circulation is further
increased the stagnation point will no longer be found on the cylinder
surface, but will appear in the flow as shown in (d) in Fig. 4.34.

The Cpdistribution is plotted in
Fig.4.35
(a) and is also shown plotted along the cylinder surface in Fig.4.35
(b). Asymmetry about x-axis is evident indicating the generation of
lift. Drag however is zero. Magnitude of the lift force is calculated by
integration as in Eqn. 4.118 and
4.119 .

The result derived above, namely, is a very general one and is valid
for any closed body placed in a uniform stream. It is named the Kutta-Joukowsky theorem in honour of Kutta and
Joukowsky who proved it independently in 1902 and 1906 respectively. The
theorem finds considerable application in calculating lift around
aerofoils. See Fig.4.37.

We have shown in Eqn.4.144
that a force is produced when circulation is imposed upon a cylinder
placed in uniform flow (see Fig. 4.38) .
This force is nothing but the lift. This effect is called Magnus Effect in honour of the scholar Heinrich Magnus (1802 - 1870). Sports involving balls,
such as golf, baseball, tennis see this effect in action. A spinning ball
when hit in a horizontal direction follows a curved trajectory because of
this effect.